Dielectric

Dielectric Materials: Behavior and Mathematical Analysis

Dielectrics are insulating materials that, when placed in an electric field, become polarized but do not conduct electricity. Their main function in capacitors and other electrostatic applications is to reduce the effective electric field and increase capacitance.

Polar vs. Non-Polar Dielectrics

Polar dielectrics have molecules with a permanent electric dipole moment, meaning their centers of positive and negative charge do not coincide. Water is a classic example.

Non-polar dielectrics have molecules with no permanent dipole. But when exposed to an external electric field, the atoms or molecules temporarily develop dipole moments due to the influence of the field.

Polarization of Dielectric

When a dielectric is placed in an external electric field, its molecules align in a way that creates an opposing internal electric field. This phenomenon is called polarization.

Polarization vector: \( \vec{P} \) represents the electric dipole moment per unit volume and reflects the extent to which the dielectric material becomes polarized.

Effective Electric Field Inside a Dielectric

The net electric field inside a dielectric is reduced due to polarization. This is mathematically expressed as:

\( E_{net} = E - E_d \)

• \( E \): Applied external electric field
• \( E_d \): Electric field produced by polarization (opposes \( E \))

This internal electric field acts to oppose the external field, thereby lowering the overall electric field within the dielectric material.

Electric Field Due to Surface Charge

In the case of a parallel plate capacitor, the electric field between plates (without dielectric) is given by:

\( E = \frac{\sigma}{\varepsilon_0} \)

• \( \sigma \): Represents the amount of electric charge per unit area on the surface of the plates
• \( \varepsilon_0 \): The vacuum permittivity, indicating the ability of free space to permit electric field lines

Introducing a dielectric with relative permittivity \( \varepsilon_r \) reduces the electric field to:

\( E = \frac{\sigma}{\varepsilon_0 \varepsilon_r} \)

Reduction in Field Due to Dielectric

The change in electric field caused by inserting the dielectric can be found as:

\( E_{air} - E_{dielectric} = \frac{\sigma}{\varepsilon_0} - \frac{\sigma}{\varepsilon_0 \varepsilon_r} = \frac{\sigma}{\varepsilon_0} \left(1 - \frac{1}{\varepsilon_r}\right) \)

This difference represents the electric field due to the polarization, i.e., \( E_d \):

\( E_d = \frac{\sigma}{\varepsilon_0} \left(1 - \frac{1}{\varepsilon_r} \right) \)

Therefore, the effective field inside the dielectric is:

\( E_{net} = E - E_d = \frac{\sigma}{\varepsilon_0 \varepsilon_r} \)

Why Water Acts as a Dielectric but Not as a Capacitor

Water is a polar dielectric, meaning it gets polarized in an electric field, but it cannot function as a capacitor alone. This is because a capacitor requires two conductors separated by an insulating medium. Water, being a liquid and conductive under certain conditions, cannot replace the entire structure of a capacitor. Instead, it serves as the insulating medium (dielectric) between conducting plates.

Stored Energy in a Capacitor Containing a Dielectric

When a dielectric is inserted into a capacitor, it increases the energy-storing capacity. The amount of energy a capacitor holds can be expressed using the following formula:

\( U = \frac{1}{2} C V^2 \)

Where:

• \( C \): Capacitance of the capacitor
• \( V \): The electric potential difference between the two plates

Since \( C = \varepsilon_0 \varepsilon_r \frac{A}{d} \), the dielectric increases the capacitance, and thus the energy stored also increases.